Theorem ltexnq 10987

Description: Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119. (Contributed by NM, 24-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
ltexnq	⊢ (𝐵 ∈ Q → (𝐴 <Q 𝐵 ↔ ∃𝑥(𝐴 +Q 𝑥) = 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ltexnq

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelnq 10938	. . . 4 ⊢ <Q ⊆ (Q × Q)
2	1	brel 5724	. . 3 ⊢ (𝐴 <Q 𝐵 → (𝐴 ∈ Q ∧ 𝐵 ∈ Q))
3		ordpinq 10955	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 ↔ ((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
4		elpqn 10937	. . . . . . . . 9 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
5	4	adantr 486	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐴 ∈ (N × N))
6		xp1st 8021	. . . . . . . 8 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
7	5, 6	syl 18	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1st ‘𝐴) ∈ N)
8		elpqn 10937	. . . . . . . . 9 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N))
9	8	adantl 487	. . . . . . . 8 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐵 ∈ (N × N))
10		xp2nd 8022	. . . . . . . 8 ⊢ (𝐵 ∈ (N × N) → (2nd ‘𝐵) ∈ N)
11	9, 10	syl 18	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘𝐵) ∈ N)
12		mulclpi 10905	. . . . . . 7 ⊢ (((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
13	7, 11, 12	syl2anc 596	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
14		xp1st 8021	. . . . . . . 8 ⊢ (𝐵 ∈ (N × N) → (1st ‘𝐵) ∈ N)
15	9, 14	syl 18	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (1st ‘𝐵) ∈ N)
16		xp2nd 8022	. . . . . . . 8 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
17	5, 16	syl 18	. . . . . . 7 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘𝐴) ∈ N)
18		mulclpi 10905	. . . . . . 7 ⊢ (((1st ‘𝐵) ∈ N ∧ (2nd ‘𝐴) ∈ N) → ((1st ‘𝐵) ·N (2nd ‘𝐴)) ∈ N)
19	15, 17, 18	syl2anc 596	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((1st ‘𝐵) ·N (2nd ‘𝐴)) ∈ N)
20		ltexpi 10914	. . . . . 6 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) ∈ N ∧ ((1st ‘𝐵) ·N (2nd ‘𝐴)) ∈ N) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) ↔ ∃𝑦 ∈ N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴))))
21	13, 19, 20	syl2anc 596	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) ↔ ∃𝑦 ∈ N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴))))
22		relxp 5677	. . . . . . . . . . . 12 ⊢ Rel (N × N)
23	4	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → 𝐴 ∈ (N × N))
24		1st2nd 8039	. . . . . . . . . . . 12 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
25	22, 23, 24	sylancr 599	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
26	25	oveq1d 7431	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩))
27	7	adantr 486	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (1st ‘𝐴) ∈ N)
28	17	adantr 486	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (2nd ‘𝐴) ∈ N)
29		simpr 490	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → 𝑦 ∈ N)
30		mulclpi 10905	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝐴) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
31	17, 11, 30	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
32	31	adantr 486	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)
33		addpipq 10949	. . . . . . . . . . 11 ⊢ ((((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐴) ∈ N) ∧ (𝑦 ∈ N ∧ ((2nd ‘𝐴) ·N (2nd ‘𝐵)) ∈ N)) → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))⟩)
34	27, 28, 29, 32, 33	syl22anc 852	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))⟩)
35	26, 34	eqtrd 2797	. . . . . . . . 9 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))⟩)
36		oveq2 7424	. . . . . . . . . . . 12 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ((2nd ‘𝐴) ·N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦)) = ((2nd ‘𝐴) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴))))
37		distrpi 10910	. . . . . . . . . . . . 13 ⊢ ((2nd ‘𝐴) ·N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦)) = (((2nd ‘𝐴) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) +N ((2nd ‘𝐴) ·N 𝑦))
38		fvex 6895	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘𝐴) ∈ V
39		fvex 6895	. . . . . . . . . . . . . . 15 ⊢ (1st ‘𝐴) ∈ V
40		fvex 6895	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘𝐵) ∈ V
41		mulcompi 10908	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
42		mulasspi 10909	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
43	38, 39, 40, 41, 42	caov12 7645	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝐴) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) = ((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))
44		mulcompi 10908	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝐴) ·N 𝑦) = (𝑦 ·N (2nd ‘𝐴))
45	43, 44	oveq12i 7428	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝐴) ·N ((1st ‘𝐴) ·N (2nd ‘𝐵))) +N ((2nd ‘𝐴) ·N 𝑦)) = (((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴)))
46	37, 45	eqtr2i 2786	. . . . . . . . . . . 12 ⊢ (((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))) = ((2nd ‘𝐴) ·N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦))
47		mulasspi 10909	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)) = ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (1st ‘𝐵)))
48		mulcompi 10908	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝐴) ·N (1st ‘𝐵)) = ((1st ‘𝐵) ·N (2nd ‘𝐴))
49	48	oveq2i 7427	. . . . . . . . . . . . 13 ⊢ ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (1st ‘𝐵))) = ((2nd ‘𝐴) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))
50	47, 49	eqtri 2785	. . . . . . . . . . . 12 ⊢ (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)) = ((2nd ‘𝐴) ·N ((1st ‘𝐵) ·N (2nd ‘𝐴)))
51	36, 46, 50	3eqtr4g 2822	. . . . . . . . . . 11 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → (((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))) = (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)))
52		mulasspi 10909	. . . . . . . . . . . . 13 ⊢ (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵)) = ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))
53	52	eqcomi 2771	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) = (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))
54	53	a1i 11	. . . . . . . . . . 11 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) = (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵)))
55	51, 54	opeq12d 4844	. . . . . . . . . 10 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))⟩ = ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩)
56	55	eqeq2d 2773	. . . . . . . . 9 ⊢ ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ((𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((1st ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵))) +N (𝑦 ·N (2nd ‘𝐴))), ((2nd ‘𝐴) ·N ((2nd ‘𝐴) ·N (2nd ‘𝐵)))⟩ ↔ (𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩))
57	35, 56	syl5ibcom 248	. . . . . . . 8 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → (𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩))
58		fveq2 6882	. . . . . . . . 9 ⊢ ((𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ → ([Q]‘(𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩))
59		adderpq 10968	. . . . . . . . . . 11 ⊢ (([Q]‘𝐴) +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = ([Q]‘(𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩))
60		nqerid 10945	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴)
61	60	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ([Q]‘𝐴) = 𝐴)
62	61	oveq1d 7431	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (([Q]‘𝐴) +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)))
63	59, 62	eqtr3id 2811	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ([Q]‘(𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)))
64		mulclpi 10905	. . . . . . . . . . . . . . . 16 ⊢ (((2nd ‘𝐴) ∈ N ∧ (2nd ‘𝐴) ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N)
65	17, 17, 64	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N)
66	65	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N)
67	15	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (1st ‘𝐵) ∈ N)
68	11	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (2nd ‘𝐵) ∈ N)
69		mulcanenq 10972	. . . . . . . . . . . . . 14 ⊢ ((((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N ∧ (1st ‘𝐵) ∈ N ∧ (2nd ‘𝐵) ∈ N) → ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ~Q ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
70	66, 67, 68, 69	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ~Q ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
71	8	ad2antlr 740	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → 𝐵 ∈ (N × N))
72		1st2nd 8039	. . . . . . . . . . . . . 14 ⊢ ((Rel (N × N) ∧ 𝐵 ∈ (N × N)) → 𝐵 = ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
73	22, 71, 72	sylancr 599	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → 𝐵 = ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩)
74	70, 73	breqtrrd 5137	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ~Q 𝐵)
75		mulclpi 10905	. . . . . . . . . . . . . . 15 ⊢ ((((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N ∧ (1st ‘𝐵) ∈ N) → (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)) ∈ N)
76	66, 67, 75	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)) ∈ N)
77		mulclpi 10905	. . . . . . . . . . . . . . 15 ⊢ ((((2nd ‘𝐴) ·N (2nd ‘𝐴)) ∈ N ∧ (2nd ‘𝐵) ∈ N) → (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵)) ∈ N)
78	66, 68, 77	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵)) ∈ N)
79	76, 78	opelxpd 5698	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ∈ (N × N))
80		nqereq 10947	. . . . . . . . . . . . 13 ⊢ ((⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ~Q 𝐵 ↔ ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩) = ([Q]‘𝐵)))
81	79, 71, 80	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ ~Q 𝐵 ↔ ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩) = ([Q]‘𝐵)))
82	74, 81	mpbid 235	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩) = ([Q]‘𝐵))
83		nqerid 10945	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ Q → ([Q]‘𝐵) = 𝐵)
84	83	ad2antlr 740	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ([Q]‘𝐵) = 𝐵)
85	82, 84	eqtrd 2797	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩) = 𝐵)
86	63, 85	eqeq12d 2778	. . . . . . . . 9 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → (([Q]‘(𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = ([Q]‘⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩) ↔ (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = 𝐵))
87	58, 86	imbitrid 247	. . . . . . . 8 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((𝐴 +pQ ⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) = ⟨(((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (1st ‘𝐵)), (((2nd ‘𝐴) ·N (2nd ‘𝐴)) ·N (2nd ‘𝐵))⟩ → (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = 𝐵))
88	57, 87	syld 48	. . . . . . 7 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = 𝐵))
89		fvex 6895	. . . . . . . 8 ⊢ ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) ∈ V
90		oveq2 7424	. . . . . . . . 9 ⊢ (𝑥 = ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) → (𝐴 +Q 𝑥) = (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)))
91	90	eqeq1d 2764	. . . . . . . 8 ⊢ (𝑥 = ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩) → ((𝐴 +Q 𝑥) = 𝐵 ↔ (𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = 𝐵))
92	89, 91	spcev 3563	. . . . . . 7 ⊢ ((𝐴 +Q ([Q]‘⟨𝑦, ((2nd ‘𝐴) ·N (2nd ‘𝐵))⟩)) = 𝐵 → ∃𝑥(𝐴 +Q 𝑥) = 𝐵)
93	88, 92	syl6 36	. . . . . 6 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑦 ∈ N) → ((((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
94	93	rexlimdva 3165	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (∃𝑦 ∈ N (((1st ‘𝐴) ·N (2nd ‘𝐵)) +N 𝑦) = ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
95	21, 94	sylbid 243	. . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (((1st ‘𝐴) ·N (2nd ‘𝐵)) <N ((1st ‘𝐵) ·N (2nd ‘𝐴)) → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
96	3, 95	sylbid 243	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q 𝐵 → ∃𝑥(𝐴 +Q 𝑥) = 𝐵))
97	2, 96	mpcom 39	. 2 ⊢ (𝐴 <Q 𝐵 → ∃𝑥(𝐴 +Q 𝑥) = 𝐵)
98		eleq1 2850	. . . . . . 7 ⊢ ((𝐴 +Q 𝑥) = 𝐵 → ((𝐴 +Q 𝑥) ∈ Q ↔ 𝐵 ∈ Q))
99	98	biimparc 485	. . . . . 6 ⊢ ((𝐵 ∈ Q ∧ (𝐴 +Q 𝑥) = 𝐵) → (𝐴 +Q 𝑥) ∈ Q)
100		addnqf 10960	. . . . . . . 8 ⊢ +Q :(Q × Q)⟶Q
101	100	fdmi 6718	. . . . . . 7 ⊢ dom +Q = (Q × Q)
102		0nnq 10936	. . . . . . 7 ⊢ ¬ ∅ ∈ Q
103	101, 102	ndmovrcl 7603	. . . . . 6 ⊢ ((𝐴 +Q 𝑥) ∈ Q → (𝐴 ∈ Q ∧ 𝑥 ∈ Q))
104		ltaddnq 10986	. . . . . 6 ⊢ ((𝐴 ∈ Q ∧ 𝑥 ∈ Q) → 𝐴 <Q (𝐴 +Q 𝑥))
105	99, 103, 104	3syl 19	. . . . 5 ⊢ ((𝐵 ∈ Q ∧ (𝐴 +Q 𝑥) = 𝐵) → 𝐴 <Q (𝐴 +Q 𝑥))
106		simpr 490	. . . . 5 ⊢ ((𝐵 ∈ Q ∧ (𝐴 +Q 𝑥) = 𝐵) → (𝐴 +Q 𝑥) = 𝐵)
107	105, 106	breqtrd 5135	. . . 4 ⊢ ((𝐵 ∈ Q ∧ (𝐴 +Q 𝑥) = 𝐵) → 𝐴 <Q 𝐵)
108	107	ex 418	. . 3 ⊢ (𝐵 ∈ Q → ((𝐴 +Q 𝑥) = 𝐵 → 𝐴 <Q 𝐵))
109	108	exlimdv 1966	. 2 ⊢ (𝐵 ∈ Q → (∃𝑥(𝐴 +Q 𝑥) = 𝐵 → 𝐴 <Q 𝐵))
110	97, 109	impbid2 229	1 ⊢ (𝐵 ∈ Q → (𝐴 <Q 𝐵 ↔ ∃𝑥(𝐴 +Q 𝑥) = 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃wrex 3088  ⟨cop 4593   class class class wbr 5107   × cxp 5657  Rel wrel 5664  ‘cfv 6537  (class class class)co 7416  1^st c1st 7987  2^nd c2nd 7988  Ncnpi 10856   +_N cpli 10857   ·_N cmi 10858   <_N clti 10859   +_pQ cplpq 10860   ~_Q ceq 10863  Qcnq 10864  [Q]cerq 10866   +_Q cplq 10867   <_Q cltq 10870

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-oadd 8462  df-omul 8463  df-er 8699  df-ni 10884  df-pli 10885  df-mi 10886  df-lti 10887  df-plpq 10920  df-mpq 10921  df-ltpq 10922  df-enq 10923  df-nq 10924  df-erq 10925  df-plq 10926  df-mq 10927  df-1nq 10928  df-ltnq 10930

This theorem is used by:  ltbtwnnq  10990  prnmadd  11009  ltexprlem4  11051  ltexprlem7  11054  prlem936  11059